Conceptually the FFT takes advantage of a shortcut similar to the distributive law for multiplication. To compute $$(x_1 + x_2)(x_3 + x_4)$$ on could either add first (twice) and then multiply (once), or one could expand $$sx_1x_3 + x_1x_4 + x_2x_3 + x_2x_4$$ and multiply (four times) and then add (three times). This idea has been spelled out in the paper The Generalized Distributive Law.
Conceptually the FFT takes advantage of a shortcut similar to the distributive law for multiplication. To compute $$(x_1 + x_2)(x_3 + x_4)$$ on could either add first (twice) and then multiply (once), or one could expand $$sx_1x_3 + x_1x_4 + x_2x_3 + x_2x_4$$ and multiply (four times) and then add (three times). This idea has been spelled out in the paper The Generalized Distributive Law.